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Functions and Their Notation

What is a function?

A function is a mathematical object that describes the relationship between an input and an output. Think of it like a box that takes in some input value $x$ and spits out an output value $f(x)$.

Example

Let $f(x) = 2x + 3$. Here, $f$ is the function name, $x$ is the input, and $2x + 3$ is the rule that defines the function.

A function performs an operation on the input to turn it into an output. For example, the function $f(x) = x^2 + 4$ squares the input, then adds 4.

It's important that the function only has one output for each input. If a single input can map to multiple outputs, for example $f(x) = \pm \sqrt{x}$, then that is not a function, simply a relation.

Note

A function can still map different inputs to a single output, for example $f(x) = x^2$ maps both $-2$ and $2$ to $4$. As long as it doesn't yield two different possible outputs for the same input, it can still be considered a function.

Function Notation

The most notation for a function is $f(x)$, where $f$ is the name of the function and $x$ is the input variable. However, other letters can be used depending on the context:

$v(t)$ might represent velocity as a function of time
$C(n)$ could denote cost as a function of the number of items

Tip

When working with multiple functions, exams may use different letters to distinguish them, such as $f(x)$, $g(x)$, and $h(x)$.

Occasionally, the following notation will also be used:

$$f : x \to y$$

where $y$ is an expression. This means the same thing as $f(x) = y$, where the function maps the input $x$ to the output $y$.

When evaluating a function at a specific number, we use the notation $f(n)$ where $n$ is the number you are evaluating the function at.

Example

Let $f(x) = \sqrt{x + 4}$. The value of $f(x)$ at $x = 5$ is $f(5) = \sqrt{4 + 5} = 3$.

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) that the function can produce.

Domain

The domain is often determined by considering restrictions on the input values:

For rational functions, exclude values that make the denominator zero.
For square root functions, ensure the expression under the square root is non-negative.
For logarithmic functions, the argument must be positive.

Example

For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except 2, because when $x = 2$, the denominator becomes zero.

For $f(x) = \sqrt{1 - x}$, the domain is all real numbers less than or equal to 1, because if $x > 1$, the expression under the square root is negative.

For $f(x) = \log(2x - 1)$, the domain is all real numbers greater than $\frac12$, because if $x \leq \frac12$, the argument is negative or zero.

For some functions, the domain can also be restricted. This means that the function is essentially "cut off" at all points not in the domain defined.

Example

$f(x) = x^2, x\geq 0$ means the function is only defined for values of $x$ greater than or equal to $0$.

Common Mistake

Students often forget to check whether the range changes as a result of a restricted domain. For example, the range of $f(x) = \cos x$ is $-1 \leq x \leq 1$, but the range of $f(x) = \cos x, -\frac\pi2 \leq x \leq \frac\pi2$ is $0 \leq x \leq 1$.

Range

The range can often be determined by analyzing the function's behavior:

For polynomial functions, consider end behavior and turning points.
For trigonometric functions, recall their periodic nature.
For exponential functions, consider asymptotic behavior.

Example

For $g(x) = x^2$, the range is all non-negative real numbers, because a square is always non-negative.

For $g(x) = \sin x$, the range is all real numbers between -1 and 1.

For $g(x) = e^x$, the range is all positive real numbers, because $e^x$ cannot equal a negative number or zero.

Notating domain and range

Domains and ranges can be written using inequalities. For example:

The domain of $f(x) = \sqrt{x}$ is $x\geq 0$.
The range of $f(x) = \sin x$ is $-1 \leq f(x) \leq 1$.

Note

Ranges can also be written using $y$, for example in the case of $f(x) = \sin x$ you can write the range as $-1 \leq y \leq 1$. Some pedantic mathematicians might yell at you, but the examiners will know what you mean.

Domains and ranges are more formally written in interval notation. Interval notation defines an interval with two numbers, a lower bound and an upper bound, with a square bracket $[$ meaning inclusive and a round bracket $($ meaning exclusive. For example:

$x \in [0, 4]$ means $0 \leq x \leq 4$
$y \in (-4, -2)$ means $-4 < y < -2$
$r \in [0, \pi)$ means $0 \leq r < \pi$

If one end is unbounded, use $\infty$ or $-\infty$, always with a round bracket. For example:

$x \in [0, \infty)$ means $x \geq 0$
$a \in (-\infty, -2)$ means $x < -2$

Common Mistake

Never, ever write $\infty$ with a square bracket (e.g. $x \in [2, \infty]$). You will not get the mark on the exam.

If there are no bounds (i.e. domain or range is all real numbers), you can simply write:

$x \in {\rm I\!R}$, meaning $x$ is an element of the real numbers

Finally, if you want to combine two intervals, you can use the union operator $\cup$. For example:

$x \in [-2, 0] \cup [2, \infty)$, meaning $-2 \leq x \leq 0$ or $x \geq 2$
$y \in (-\infty, 1) \cup (1, \infty)$, meaning $y \neq 1$

Note

An alternative notation for intervals is using a square bracket facing the wrong way for an open interval. For example:

$x \in [0, 2[$ meaning $0 \leq x < 2$
$y \in\ ]0, \infty[$ meaning $y > 0$

These notes will mostly use round brackets, but the two notations are interchangeable, so use whichever one you prefer.

Functions as Mathematical Models

Functions serve as powerful tools for modeling real-world phenomena. They allow us to describe relationships between variables and make predictions based on these relationships.

Example

The function $h(t) = -4.9t^2 + 20t + 1.5$ could model the height (in meters) of a ball thrown upwards after $t$ seconds, where 1.5 meters is the initial height, 20 m/s is the initial velocity, and -4.9 accounts for gravity.

Note

When using functions as models, it's crucial to consider the domain and range in the context of the real-world situation. For instance, in the ball-throwing example, time ($t$) cannot be negative, and the height will eventually become zero when the ball hits the ground.

Inverse Functions

An inverse function "undoes" what the original function does. If a function $f$ takes $x$ to $y$, then its inverse $f^{-1}$ takes $y$ back to $x$.

Notation

The inverse of a function $f$ is denoted as $f^{-1}$. It's important to note that this is not the same as $\frac{1}{f}$.

Common Mistake

Students often confuse $f^{-1}(x)$ with $\frac{1}{f(x)}$. Remember, $f^{-1}(x)$ is the inverse function, while $\frac{1}{f(x)}$ is the reciprocal of the function value.

Graphical Representation

Graphically, the inverse function is a reflection of the original function over the line $y = x$. This means that for every point $(a, b)$ on the graph of $f$, there is a corresponding point $(b, a)$ on the graph of $f^{-1}$.

Note

Not all functions have inverses. For a function to have an inverse, it must be one-to-one (injective), meaning each element in the range corresponds to at most one element in the domain.

Finding Inverse Functions

To find the inverse of a function algebraically:

Replace $f(x)$ with $y$
Swap $x$ and $y$
Solve for $y$
Replace $y$ with $f^{-1}(x)$

Example

Let's find the inverse of $f(x) = 2x + 3$:

$y = 2x + 3$
$x = 2y + 3$
$x - 3 = 2y$ $\frac{x - 3}{2} = y$
$f^{-1}(x) = \frac{x - 3}{2}$

Tip

To check if you've found the correct inverse, compose the original function with the inverse. The result should be $x$:

$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

Conclusion

Understanding functions, their notation, domain, range, and inverses is crucial in IB Mathematics AA SL. These concepts form the foundation for more advanced topics in calculus and mathematical modeling. By mastering these ideas, students can develop a deeper appreciation for the power and versatility of functions in describing real-world phenomena and solving complex problems.

Functions and Their Notation

What is a function?

A function is a mathematical object that describes the relationship between an input and an output. Think of it like a box that takes in some input value $x$ and spits out an output value $f(x)$.

Example

Let $f(x) = 2x + 3$. Here, $f$ is the function name, $x$ is the input, and $2x + 3$ is the rule that defines the function.

A function performs an operation on the input to turn it into an output. For example, the function $f(x) = x^2 + 4$ squares the input, then adds 4.

It's important that the function only has one output for each input. If a single input can map to multiple outputs, for example $f(x) = \pm \sqrt{x}$, then that is not a function, simply a relation.

Note

Function Notation

The most notation for a function is $f(x)$, where $f$ is the name of the function and $x$ is the input variable. However, other letters can be used depending on the context:

$v(t)$ might represent velocity as a function of time
$C(n)$ could denote cost as a function of the number of items

Tip

When working with multiple functions, exams may use different letters to distinguish them, such as $f(x)$, $g(x)$, and $h(x)$.

Occasionally, the following notation will also be used:

$$f : x \to y$$

where $y$ is an expression. This means the same thing as $f(x) = y$, where the function maps the input $x$ to the output $y$.

When evaluating a function at a specific number, we use the notation $f(n)$ where $n$ is the number you are evaluating the function at.

Example

Let $f(x) = \sqrt{x + 4}$. The value of $f(x)$ at $x = 5$ is $f(5) = \sqrt{4 + 5} = 3$.

Domain and Range

Domain

The domain is often determined by considering restrictions on the input values:

For rational functions, exclude values that make the denominator zero.
For square root functions, ensure the expression under the square root is non-negative.
For logarithmic functions, the argument must be positive.

Example

For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except 2, because when $x = 2$, the denominator becomes zero.

For $f(x) = \sqrt{1 - x}$, the domain is all real numbers less than or equal to 1, because if $x > 1$, the expression under the square root is negative.

For $f(x) = \log(2x - 1)$, the domain is all real numbers greater than $\frac12$, because if $x \leq \frac12$, the argument is negative or zero.

For some functions, the domain can also be restricted. This means that the function is essentially "cut off" at all points not in the domain defined.

Example

$f(x) = x^2, x\geq 0$ means the function is only defined for values of $x$ greater than or equal to $0$.

Common Mistake

Range

The range can often be determined by analyzing the function's behavior:

For polynomial functions, consider end behavior and turning points.
For trigonometric functions, recall their periodic nature.
For exponential functions, consider asymptotic behavior.

Example

For $g(x) = x^2$, the range is all non-negative real numbers, because a square is always non-negative.

For $g(x) = \sin x$, the range is all real numbers between -1 and 1.

For $g(x) = e^x$, the range is all positive real numbers, because $e^x$ cannot equal a negative number or zero.

Notating domain and range

Domains and ranges can be written using inequalities. For example:

The domain of $f(x) = \sqrt{x}$ is $x\geq 0$.
The range of $f(x) = \sin x$ is $-1 \leq f(x) \leq 1$.

Note

$x \in [0, 4]$ means $0 \leq x \leq 4$
$y \in (-4, -2)$ means $-4 < y < -2$
$r \in [0, \pi)$ means $0 \leq r < \pi$

If one end is unbounded, use $\infty$ or $-\infty$, always with a round bracket. For example:

$x \in [0, \infty)$ means $x \geq 0$
$a \in (-\infty, -2)$ means $x < -2$

Common Mistake

Never, ever write $\infty$ with a square bracket (e.g. $x \in [2, \infty]$). You will not get the mark on the exam.

If there are no bounds (i.e. domain or range is all real numbers), you can simply write:

$x \in {\rm I\!R}$, meaning $x$ is an element of the real numbers

Finally, if you want to combine two intervals, you can use the union operator $\cup$. For example:

$x \in [-2, 0] \cup [2, \infty)$, meaning $-2 \leq x \leq 0$ or $x \geq 2$
$y \in (-\infty, 1) \cup (1, \infty)$, meaning $y \neq 1$

Note

An alternative notation for intervals is using a square bracket facing the wrong way for an open interval. For example:

$x \in [0, 2[$ meaning $0 \leq x < 2$
$y \in\ ]0, \infty[$ meaning $y > 0$

These notes will mostly use round brackets, but the two notations are interchangeable, so use whichever one you prefer.

Functions as Mathematical Models

Functions serve as powerful tools for modeling real-world phenomena. They allow us to describe relationships between variables and make predictions based on these relationships.

Example

Note

Inverse Functions

An inverse function "undoes" what the original function does. If a function $f$ takes $x$ to $y$, then its inverse $f^{-1}$ takes $y$ back to $x$.

Notation

The inverse of a function $f$ is denoted as $f^{-1}$. It's important to note that this is not the same as $\frac{1}{f}$.

Common Mistake

Students often confuse $f^{-1}(x)$ with $\frac{1}{f(x)}$. Remember, $f^{-1}(x)$ is the inverse function, while $\frac{1}{f(x)}$ is the reciprocal of the function value.

Graphical Representation

Note

Not all functions have inverses. For a function to have an inverse, it must be one-to-one (injective), meaning each element in the range corresponds to at most one element in the domain.

Finding Inverse Functions

To find the inverse of a function algebraically:

Replace $f(x)$ with $y$
Swap $x$ and $y$
Solve for $y$
Replace $y$ with $f^{-1}(x)$

Example

Let's find the inverse of $f(x) = 2x + 3$:

$y = 2x + 3$
$x = 2y + 3$
$x - 3 = 2y$ $\frac{x - 3}{2} = y$
$f^{-1}(x) = \frac{x - 3}{2}$

Tip

To check if you've found the correct inverse, compose the original function with the inverse. The result should be $x$:

$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

Conclusion

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Tip

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SL 2.2—Functions, domains, range, graphs Notes

Functions and Their Notation

What is a function?

Function Notation

Domain and Range

Domain

Range

Notating domain and range

Functions as Mathematical Models

Inverse Functions

Notation

Graphical Representation

Finding Inverse Functions

Conclusion

Function: Definition and Basic Understanding

Functions and Their Notation

What is a function?

Function Notation

Domain and Range

Domain

Range

Notating domain and range

Functions as Mathematical Models

Inverse Functions

Notation

Graphical Representation

Finding Inverse Functions

Conclusion

Unlock the rest of this chapter with a Free account

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Function: Definition and Basic Understanding

Unlock the rest of this chapter with a Free account

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Duis aute irure dolor in reprehenderit

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Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

SL 2.2—Functions, domains, range, graphs Notes

Functions and Their Notation

What is a function?

Function Notation

Domain and Range

Domain

Range

Notating domain and range

Functions as Mathematical Models

Inverse Functions

Notation

Graphical Representation

Finding Inverse Functions

Conclusion

Function: Definition and Basic Understanding

Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

Functions and Their Notation

What is a function?

Function Notation

Domain and Range

Domain

Range

Notating domain and range

Functions as Mathematical Models

Inverse Functions

Notation

Graphical Representation

Finding Inverse Functions

Conclusion

Unlock the rest of this chapter with a Free account

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Function: Definition and Basic Understanding

Unlock the rest of this chapter with a Free account

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident